where the lower bound is the wealth attained by a mixture portfolio in the previous section.
Here, the equality holds if and only if \(\widetilde{c}^t\in \lbrace 0,1\rbrace ^t\) .
Hereafter, we call \(\mathsf {Wealth}_t^{\pi }(\widetilde{c}^t;o_1,o_2)\)  the \(\pi \) -mixture wealth, or equivalently, the wealth of the \(\pi \) -mixture portfolio.
We remark that
[1]}'s universal portfolio (UP) is a \(\mathrm {Beta}(b;\alpha ,\beta )\) -mixture portfolio; note that \(\alpha =\beta =\frac{1}{2}\)  or \(\alpha =\beta =1\)  are typical choices due to the simplicity and the minimax optimality, respectively [1]}, [3]}.
Note that a beta distribution is a natural choice for an ease of implementation, since it is a conjugate prior of the summand \(b^k(1-b)^{t-k}\) , which can be understood as the (unnormalized) binomial distribution.
